Find the area of the region bounded by the curves y 2 = 4ax and x 2 = 4ay
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y² = 4 a x and y = x² / 4 a
find intersection points: y² = 4 a x = x⁴ /16a²
=> x,y = (0, 0) or (4 a, 4a)
Area:
![= \int\limits^a_b {[y_1-y_2]} \, dx \\\\= \int\limits^{4a}_0 {[\sqrt{4a}\sqrt{x}-x^2/4a]} \, dx \\\\=[\frac{4\sqrt{a}}{3}(x)^\frac{3}{2}-\frac{x^3}{12a}]_0^{4a}\\\\=\frac{32a^2}{3}-\frac{16a^2}{3}\\\\=\frac{16a^2}{3}](https://tex.z-dn.net/?f=%3D+%5Cint%5Climits%5Ea_b+%7B%5By_1-y_2%5D%7D+%5C%2C+dx+%5C%5C%5C%5C%3D+%5Cint%5Climits%5E%7B4a%7D_0+%7B%5B%5Csqrt%7B4a%7D%5Csqrt%7Bx%7D-x%5E2%2F4a%5D%7D+%5C%2C+dx+%5C%5C%5C%5C%3D%5B%5Cfrac%7B4%5Csqrt%7Ba%7D%7D%7B3%7D%28x%29%5E%5Cfrac%7B3%7D%7B2%7D-%5Cfrac%7Bx%5E3%7D%7B12a%7D%5D_0%5E%7B4a%7D%5C%5C%5C%5C%3D%5Cfrac%7B32a%5E2%7D%7B3%7D-%5Cfrac%7B16a%5E2%7D%7B3%7D%5C%5C%5C%5C%3D%5Cfrac%7B16a%5E2%7D%7B3%7D "= \int\limits^a_b {[y_1-y_2]} \, dx \\\\= \int\limits^{4a}_0 {[\sqrt{4a}\sqrt{x}-x^2/4a]} \, dx \\\\=[\frac{4\sqrt{a}}{3}(x)^\frac{3}{2}-\frac{x^3}{12a}]_0^{4a}\\\\=\frac{32a^2}{3}-\frac{16a^2}{3}\\\\=\frac{16a^2}{3}")
find intersection points: y² = 4 a x = x⁴ /16a²
=> x,y = (0, 0) or (4 a, 4a)
Area:
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